# MST124 BOOK A UNIT 1

- Created by: Scar.Rose.1
- Created on: 31-05-22 17:39

## 3

**Strategy**

To find the positive factor pairs of a positive integer

- try dividing the integer by each of the numbers 1, 2, 3, 4,... in turn

- whenever you find a factor, write it down along with the other factor in the factor pair

- stop when you get a factor pair that you have already

**The fundamental theorem of arithmetic **Every integer greater than 1 can be written as a product of prime factors in just one way

- except that you can change the order of the factors

**Strategy**

To find the prime factorisation of an integer greater than 1

- repeatedly 'factor out' the prime 2 until you obtain a number that isn't divisible by two

- repeatedly factor out the prime 3 until you obtain a number that isn't divisible by three

- repeatedly factor out the prime 5 until you obtain a number that isn't divisible by five

- continue this process with each prime number in turn, and stop when you have a produce of primes

- write out the prime factorisation with the numbers in increasing order, using index notation

## 4

**Strategy**To find the lowest common multiple or highest common factor of two or more integers greater than 1

- find the prime factorisations of the numbers

- to find the LCM, multiply together the highest power of each prime factor occurring in any of the numbers

- the find the HCF, multiply together the lowest power of each prime factor common to all the numbers

(page 23 book A)

## 5

**Strategy**

To simplify a term

- find the overall sign and write it at the front

- simplify the rest of the coefficient and write it next

- write any remaining parts of the term in an appropriate order

**Strategy**

To simplify an expression with more than one term

- identify the terms

- each term after the first starts with a plus or minus sign that isn't inside brackets

- simplify each term, including the sign at the start of each term

- collect any like terms

**Strategy**

To multiply out brackets

- multiply each term inside the brackets by the multiplier

## 6

**Strategy**

To remove brackets with a plus or minus sign in front

- if the sign is plus, keep the sign of each term inside the brackets the same

- if the sign is minus, change the sign of each term inside the brackets

- for any expressions A and B, the negative of A-B is B-A

- multiplying an expression that contains several terms by a second expression is the same as multiplying each term of the first expression by the second expression

- dividing an expression that contains several terms by a second expression is the same as dividing each term of the first expression by the second expression

Strategy

To multiply out brackets in an expression with more than one term

- identify the terms

- multiply out the brackets in each term, including the sign at the start of each resulting term

- collect any like terms

## 7

**Difference of two squares**

- for any expressions A and B

(A+B)(A-B) = A^{2}^{ }- B^{2}

(A+B)^{2} is not equal to A^{2}^{ }+ B^{2 }

**Squaring brackets **(A+B)

^{2}= A

^{2}+ 2AB + B

^{2}

(A-B)

^{2}= A

^{2}

^{ }- 2AB + B

^{2}

**Strategy**To take out a common factor from an expression

- Find a common factor of the terms (usually the HCF)

- write the common factor in front of a pair of brackets

- write what is left of each term inside the brackets

## 8

**Strategy**

To add or subtract algebraic fractions

- make sure that the fractions have the same denominator

- if necessary, rewrite each fraction as an equivalent fraction to achieve this

- add or subtract the numerators

- simplify if possible

**Strategy**

To multiply or divide algebraic fractions

- to multiply two or more algebraic fractions, multiply the numerators together and multiply the denominators together

- to divide by an algebraic fraction, multiply by its reciprocal

- simplify if possible

- for every even natural number n, every positive number has exactly two real nth roots

- one positive one negative

- every negative number has no real nth roots

- the number 0 has 1 nth root - itself

- for every odd natural number n, every real number has exactly one real nth root

## 9

- the square root of a product of numbers is the same as the product of the square roots of the numbers

- the same rule applies to quotients

√ab = √a√b

√(a/b) = √a/√b

- you can rationalize the denominator of the surd a/√b by multiplying the top and bottom by √b

- when the denominator is a sum of two terms, either or both of which is a rational number multiplies by an irrational square root, rationalise the denominator by multiplying the top and bottom of the fraction by a conjugate of the expression in the denominator

- this is an expression that is obtained by changing the sign of one of the two terms (generally the second term)

## 10

**Index laws for a single base**

-to multiply two powers with the same base, add the indices

a^{m}a^{n} = a^{m+n}- to divide two powers with the same base, subtract the indices

a^{m}/a^{n} = a^{m-n}

- to find a power of a power, multiply the indices

(a^{m})^{n} = a^{mn}- a number raised to the power 0 is 1

a^{0 }= 1

- a number raised to a negative power is the reciprocal of the number raised to the corresponding positive power

a^{-n} = 1/a^{n}

## 11

**Index laws for two bases**

- a power of a product of numbers is the same as the product of the same powers of the numbers

- this rule applies for a power of a quotients

(ab)^{n} = a^{n}b^{n}

(a/b)^{n} = a^{n}/b^{n}

a^{-n} = 1/a^{n}

1/a^{-n }= a^{n}

**Converting between fractional indices and roots**a

^{1/n}=

^{n}√a

a

^{m/n}= (

^{n}√a)

^{m}=

^{n}√(a

^{m})

**Scientific Notation**

To express a number in scientific notation, write it in the form:

(a number between 1 and 10, but not including 10) x (an integer power of ten)

e.g., 427 = 4.27 x 100 = 4.27 x 10^{2}

## 12

**Rearranging equations**Carrying out any of the following operations on an equation gives an equivalent equation

- rearrange the expressions on one or both sides

- swap the sides

- do the same thing to both sides

**Doing the same thing to both sides of an equation**Doing any of the following things to both sides of an equation gives an equivalent equation

- add something

- subtract something

- multiply by something (provided that it is non-zero)

- divide by something (provided that it is non-zero)

- raise to a power (provided that the power is non-zero, and that the expression on each side of the equation can take only non-negative values)

- moving a term of one side of an equation to the other side, and changing its sign gives an equivalent equation

(change the side, change the sign)

## 13

- multiplying each term on both sides of an equation by something, provided that it is non-zero, gives an equivalent equation

- dividing each term on both sides of an equation by something, once again providing its non-zero, gives an equivalent equation

**Cross Multiplying**

If A, B, C, and D are any expressions, then the equations:

(A/B) = (C/D) and AD = BC

are equivalent (provided that B and D are never zero)

**Strategy**To solve a linear equation in one unknown

- clear any fractions and multiply out any brackets

- to clear fractions, multiply through by a suitable expression

- add or subtract terms on both sides to get all the terms in the unknown on one side, and all the other terms on the other side

- collect like terms

- divide both sides by the coefficient of the unknown

## 14

**S****trategy**

To make a variable the subject of an equation (this works for some equations but not all)

- clear any fractions and multiply out any brackets

- to clear fractions, multiply through by a suitable expression

- add or subtract terms on both sides to get all the terms containing the required subject on one side, and all the other terms on the other side

- collect like terms

- if more than one term contains the required subject, then take it out as a common factor

- divide both sides by the expression that multiplies the required subject

- this strategy works provided that the equation is 'linear in the required subject'

- its form must be such that if you replace every variable other than the required subject but a suitable number (one that doesn't lead to multiplication or division by 0) then the result is a linear equation in the required subject

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